Stratos Prassidis

Groups with infinite virtual cohomological dimension which act freely on Rm × Sn − 1

In this paper free and proper discontinuous actions on Rm × Sn − 1 are constructed of groups with infinite virtual cohomological dimension.

Rigidity of Coxeter groups

Let W be a Coxeter group acting properly discontinuously and cocompactly on manifolds N and M (9M = 0) such that the fixed point sets of finite subgroups are contractible. Let f : (N, &N) -* (Mx Dk, M XSk1) be a W-homotopy equivalence which restricts to a W-homeomorphism on the boundary. Under an assumption on the three dimensional fixed point sets, we show that then f is W-homotopic to a W-homeomorphism.

On the vanishing of certain K-theory Nil groups

Let Γ i , i = 0, 1, be two groups containing C p , the cyclic group of prime order p, as a subgroup of index 2. Let Γ = Γ0 * Cp Γ1. We show that the Nil-groups appearing in Waldhausen’s splitting theorem for computing K j (ℤΓ) (j≤ 1) vanish. Thus, in low degrees, the K-theory of ℤΓ can be computed by a Mayer-Vietoris type exact sequence involving the K-theory of the integral group rings of the groups Γ0, Γ1 and C p .

Control and relaxation over the circle

Introduction and statement of results Moduli spaces of manifolds and maps Wrapping-up and unwrapping as simplicial maps Relaxation as a simplicial map The Whitehead spaces Torsion and a higher sum theorem Nil as a geometrically defined simplicial set Transfers Completion of the proof Comparison with the lower algebraic nil groups Appendix A. Controlled homotopies on mapping tori Bibliography.

On the Exponent of the

It is known that the K-theory of a large class of groups can be computed from the Ktheory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the K-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of NK0-groups that appear in the calculation of the K0-groups of virtually infinite cyclic groups. 1991 Mathematics Subject Classification 18F25, 19A31

{\nk}_0

Abstract. We show that the holomorph of the free group on two generators satisfies the Farrell–Jones Fibered Isomorphism Conjecture. As a consequence, we show that the lower K-theory of the above group vanishes.

-Groups of Virtually Infinite Cyclic Groups

We compute the equivariant zeta function for bundles over infinite graphs and for infinite covers. In particular, we give a ‘transfer formula’ for the zeta function of infinite graph covers. Also, when the infinite cover is given as a limit of finite covers, we give a formula for the limit of the zeta functions.

On the vanishing of the lower K-theory of the holomorph of a free group on two generators

The theory of approximate fibrations is extended to an equivariant setting. Equivariant approximate fibrations are characterized by considering the maps on the fixed point sets. The theory is applied to equivariant fibrations over the circle. 1991 Mathematics Subject Classification: 55R91, 57S17. Introduction Let G be a finite group. In this paper we developed the analogue of the theory of approximate fibrations with a G-action, and we prove certain important properties of the construction. We shall generalize to the equivariant setting the notion of an approximate fibration, s developed in the Appendix of [25]. Their treatment is more appropriate for categorical constructions when the base space does not have reasonable topological properties than the classical one given in [10], [11]. We call a G-map p : E -> B an approximate G-fibration if given a lifting problem ^x{0} _4 E l V Xxl Λ Β there is a controlled G-map F: Xx [0,1] -» E, i.e. a G-map F:ATx[0, l ]x[0, l ) -> £x[0,l) from φ to p which restricts to the map/x id[0il) on X* {0}. A G-map satisfies the approximate G-homotopy lifting property if it satisfies the approximate homotopy

Zeta functions of infinite graph bundles

In the first part of this paper, a geometric definition of the K-theory equivariant nilpotent groups is given. For a finite group G, the Nil-groups are defined as functors from the category of G-spaces and G-homotopy classes of G-maps to Abelian groups. In the nonequivariant case, these groups are isomorphic to the classical algebraic Nil-groups. In the second part, the Bass-Heller-Swan formula is proved for the equivafiant topological Whitehead group. The main result of this work is that if X is a compact G-ANR and G acts trivially on S 1, then Top Top ~ Top Who (X x S 1) ~ Who (X) @ K0o (X) �9 ~ilo(X) @ l~ildX).